Efficient algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the curse of dimensionality. We extend the forward-backward stochastic neural networks (FBSNNs) which depends on forward-backward stochastic differential equation (FBSDE) to solve incompressible Navier-Stokes equation. For Cahn-Hilliard equation, we derive a modified Cahn-Hilliard equation from a widely used stabilized scheme for original Cahn-Hilliard equation. This equation can be written as a continuous parabolic system, where FBSDE can be applied and the unknown solution is approximated by neural network. Also our method is successfully developed to Cahn-Hilliard-Navier-Stokes (CHNS) equation. The accuracy and stability of our methods are shown in many numerical experiments, specially in high dimension.

翻译：长期以来，由于维数灾难，求解高维偏微分方程（PDEs）的高效算法一直是一项极其困难的任务。我们扩展了基于前向-后向随机微分方程（FBSDE）的前向-后向随机神经网络（FBSNNs），以求解不可压缩Navier-Stokes方程。针对Cahn-Hilliard方程，我们从原始Cahn-Hilliard方程广泛使用的稳定化方案出发，推导出一个修正的Cahn-Hilliard方程。该方程可写为一个连续抛物型系统，其中可应用FBSDE，且未知解通过神经网络近似。此外，我们的方法成功拓展到了Cahn-Hilliard-Navier-Stokes（CHNS）方程。大量数值实验，特别是在高维情形下，展示了我们方法的精度和稳定性。